#373532

order	512
nilpotency class	2
isoclinism	isoclinic class of #16249
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂²×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, Frattini, cosocle	C₂³×C₄	C₂⁴
commutator, socle, prime radical	C₂⁴	C₂³×C₄
LCS	G ▷ C₂⁴ ▷ C₁
UCS	C₁ ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

850 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 40
transitive degree	μ_t(G) = 512

A minimal faithful permutation representation — 4 generators on 40 points:

(1,2)(3,6)(4,5)(7,8)(9,10)(11,16)(12,14)(13,15)(17,18)(19,23)(20,22)(21,24)(25,26,30,27,29,32,37,34)(28,31,36,33,35,38,40,39);(2,5)(6,8)(9,11,13,12)(10,14,15,16)(17,19,20,24)(18,21,22,23)(25,27,30,32,29,34,37,26)(28,33,36,38,35,39,40,31);(1,3,4,7)(2,6,5,8)(10,15)(14,16)(18,22)(19,24)(25,28,29,35)(26,31,32,38)(27,33,34,39)(30,36,37,40);(2,5)(6,8)(9,12,13,11)(10,16,15,14)(18,22)(21,23)(28,35)(31,38)(33,39)(36,40)

Polycyclic presentation:

2^9|a^2=h,b^2=fh,c^2=ef,d^2=fg,h^2=i,b^a=be,c^a=cf,d^a=dei,c^b=cg,d^c=di

Conjugacy classes & character table

104 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 280728);